Adiabatic/Reversible application of external field to Ideal Gas

[a_h]

Hello Fellow Physicists!

I'm having trouble with a problem from a graduate course in Statistical Mechanics. It's a two part question; I've got the first part, it's the second part that's giving me trouble.

The problem is about reversible/adiabatic processes, thinking of "reversible" as meaning "constant entropy." In part a (which I got, but it may help to have it here), we need to show that the evolution of a classical ideal gas obeys

$$TV^{\frac{2}{3}}=\mbox{constant.}$$

I did this by using $$S=-\frac{\partial F}{\partial T}$$ with $$F=U-\mu N, U=\frac{3}{2}NT$$. To get $$\frac{\partial \mu}{\partial T}$$, I solved

$$e^{\frac{\mu}{T}}=n\lambda^{3}=\frac{N}{V} \frac{h^{3}}{(2\pi mT)^{3/2}}$$

for mu. ($$\lambda$$ is the thermodynamic wavelength, n is the number density of the gas.)

Now for part b. We have a classical ideal gas in a cylinder. It is in equilibrium at a temperature $$T_{0}$$. Now we slowly (adiabatically and reversibly) apply a uniform external field to the container, in a direction along its axis of symmetry. When we are done, there is a potential in our container given by

$$u(z)=fz$$

where z is the distance along that symmetry axis and f is a constant. The question is, what is the temperature of the gas at the end of the process?

I would think that we should use the same assumptions:

$$S=-\frac{\partial F}{\partial T}=\mbox{constant}$$

and $$F=U-\mu N$$, but I don't know how u(z) gets incorporated into U. Any ideas?

Thanks for all of your time, everyone.

 

[Almsco]

Hi!

I'm not sure how to incorporate u(z) into the equation for U either. Have you tried looking at any other problems similar to this one to see if you can find any clues? Also, have you looked at the material you've been studying for this course to see if there are any hints? I hope someone else on the forum can provide some insight!